Key Points

A matrix is a rectangular array of numbers or functions, called elements, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has an order of $m \times n$.

Two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ are equal if they have the same order and their corresponding elements are equal. This means $a_{ij} = b_{ij}$ for all values of $i$ and $j$.

Key types include: Column Matrix (one column), Row Matrix (one row), Square Matrix (number of rows equals number of columns), and Zero or Null Matrix (all elements are zero).

A square matrix is a Diagonal Matrix if all its non-diagonal elements are zero. A diagonal matrix is a Scalar Matrix if all its diagonal elements are equal. A square matrix is an Identity Matrix, denoted by $I$, if all its diagonal elements are 1 and all other elements are 0.

The sum of two matrices of the same order is a matrix obtained by adding their corresponding elements. If $A = [a_{ij}]$ and $B = [b_{ij}]$, then $A+B = [a_{ij} + b_{ij}]$. Matrix addition is commutative and associative.

To multiply a matrix $A = [a_{ij}]$ by a scalar constant $k$, we multiply every element of $A$ by $k$. The resulting matrix is $kA = [k \cdot a_{ij}]$.

The product of two matrices $A$ and $B$, denoted $AB$, is defined only if the number of columns in matrix $A$ is equal to the number of rows in matrix $B$.

If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, their product $AB$ is an $m \times p$ matrix $C = [c_{ik}]$, where $c_{ik} = \sum_{j=1}^{n} a_{ij}b_{jk}$. This is the sum of products of elements of the $i$-th row of $A$ with the $k$-th column of $B$.

Matrix multiplication is associative, so $(AB)C = A(BC)$. It is distributive, so $A(B+C) = AB+AC$. However, it is not commutative in general, so $AB \neq BA$. The product of two non-zero matrices can be a zero matrix.

The transpose of a matrix $A$, denoted by $A'$ or $A^T$, is obtained by interchanging its rows and columns. If the order of $A$ is $m \times n$, the order of $A'$ is $n \times m$.

Important properties of the transpose are: $(A')' = A$, $(kA)' = kA'$, $(A+B)' = A' + B'$, and the reversal law for multiplication $(AB)' = B'A'$.

A square matrix $A$ is called symmetric if its transpose is equal to the matrix itself, which means $A' = A$. In a symmetric matrix, $a_{ij} = a_{ji}$ for all $i$ and $j$.

A square matrix $A$ is called skew-symmetric if its transpose is equal to its negative, which means $A' = -A$. In a skew-symmetric matrix, $a_{ij} = -a_{ji}$ for all $i, j$, and all diagonal elements are zero.

Any square matrix $A$ can be expressed as the sum of a symmetric matrix $P$ and a skew-symmetric matrix $Q$, where $P = \frac{1}{2}(A + A')$ and $Q = \frac{1}{2}(A - A')$.

A square matrix $A$ is invertible if there exists a square matrix $B$ of the same order such that $AB = BA = I$, where $I$ is the identity matrix. The matrix $B$ is called the inverse of $A$ and is denoted by $A^{-1}$.

The inverse of a square matrix, if it exists, is unique. If $A$ and $B$ are invertible matrices of the same order, then $(AB)^{-1} = B^{-1}A^{-1}$. This is known as the reversal law for inverses.